Throttling processes equivalent to full throttling on trees

Abstract

Consider a discrete-time process on a graph $G$ where a set $B$ of initial vertices are chosen to be colored blue (the remainder being white) and then a time step consists of every currently blue vertex forcing all of its neighbors to become blue; this process stops when every vertex of the graph is blue, and the process is called full forcing. The full throttling number of $G$ is then defined to be the minimum sum of the cardinality of $B$ and the number of time steps needed to complete the forcing process. On trees, the full throttling number is equivalent to the throttling numbers of several other graph processes, such as positive-semidefinite zero forcing, the game of cops and robbers, and the distance domination number (alternately, the $k$-radius) of a graph. For all of these, it is known that maximum possible throttling number for a tree on $n$ vertices is somewhere between $1.4502\sqrt{n}$ and $\frac{\sqrt{14}}{2}\sqrt{n}$, with the former exhibited by a family of spiders. After introducing some new ideas and methods for working with throttling on trees, this paper determines the exact full throttling number of all balanced spiders (trees with equal-length paths extending from a center vertex), and proves that their full throttling numbers are bounded above by that of paths of the same order $n$, which are known to have full throttling number $\lceil\sqrt{2n}-\frac{1}{2}\rceil$.